End-of-Unit Assessment

Answer:

B, D

Lines $AC$ and $BD$ intersect at $E$.

What is the measure of angle $BEC$?

Answer:

$108^\circ$

Clare created Figure A. Then she created Figure B by translating Triangle C and then translating Triangle D.

Select all the true statements.

Answer:

C, D

Describe a sequence of transformations that shows that Polygon A is congruent to Polygon B.

Answer:

Answers vary. Sample response: If Polygon A reflected across line $\ell$, a horizontal line 1 unit above the top of Polygon A, and then translated 3 units to the right, it matches up perfectly with Polygon B.

Minimal Tier 1 response:

• Work is complete and correct.
• Sample: Reflect across horizontal line $\ell$, 1 unit above Polygon A, and then translate 3 units right.

Tier 2 response:

• Work shows general conceptual understanding and mastery, with some errors.
• Sample errors: Drawing showing the intermediate transformation (the green polygon in the sample response), but no verbal descriptions; incomplete verbal descriptions (such as reference to a reflection without specifying a line); sequence of transformations contains a small, easily identifiable error (such as identifying the wrong number of units in the translation); sequence of transformations is correct but does not use proper vocabulary (“turn” instead of rotate; “move” instead of translate).

Tier 3 response:

• Significant errors in work demonstrate lack of conceptual understanding or mastery.
• Sample errors: explanation without reference to rigid transformations, such as “the shapes have the same lengths and angles”; a sequence of transformations that does not take Polygon A to Polygon B (with no obvious small mistakes responsible for this error); descriptions of transformations are unclear and the intended meaning is not evident.

For each pair of shapes, decide whether or not Shape A is congruent to Shape B. Explain your reasoning.

1. First pair:

2. Second pair:

 

Answer:

1. Not congruent. The shapes look congruent, but when Shape A is moved on top of Shape B with a translation and a 90-degree clockwise rotation, they do not match up.
2. Congruent. If Shape A is translated so the upper right vertex coincides with the lower left vertex of B, reflected along the horizontal side of the image of Shape A, and then rotated 90 degrees clockwise around the upper right vertex, it will be placed on top of Shape B.

Minimal Tier 1 response:

• Work is complete and correct.
• Acceptable errors: Omitting reference to lines of reflection, centers of rotation, angles of rotation, and distance of translation, provided the visual makes these things clear.
• Sample:

1. (with accompanying accurate drawing) Not congruent, because when I translate Shape A and then rotate it, it still doesn’t match up with Shape B. Alternate response: I measured the angles, and they are not the same in the two shapes.
2. (with accompanying accurate drawing) Congruent, because I can translate Shape A, reflect it, and then translate it onto Shape B.

Tier 2 response:

• Work shows general conceptual understanding and mastery, with some errors.
• Sample errors: Transformations are shown, but with no written descriptions; in part b, transformations are done mostly correctly but enough accuracy was lost that the shapes appear to coincide.

Tier 3 response:

• Significant errors in work demonstrate lack of conceptual understanding or mastery.
• Sample errors: Work states that shapes are or are not congruent with no justification; descriptions of transformations are unclear and the intended meaning is not evident; vague explanations such as “the shapes look the same.”

Lines $AD$ and $BC$ are parallel. Find the measures of the three angles in triangle $ADE$.

Answer:

Angle $ADE$: $45^\circ$; Angle $DAE$: $30^\circ$; Angle $DEA$: $105^\circ$

To create this diagram, triangle $ABC$ was translated so that $A$ goes to $C$. Then, triangle $ABC$ was translated so that $A$ goes to $B$. The measure of angle $A$ is $45^\circ$ and the measure of angle $D$ is $63^\circ$.

1. Identify at least two pairs of congruent angles in the figure and explain how you know they are congruent.

2. What is the measure of angle $CBE$? Explain how you know.

3. Name a triangle congruent to triangle $CBE$. Explain how you know.

Answer:

1. Answers vary. Any pair of corresponding angles will work. A sequence of rigid motions defines a congruence. Triangle $ABC$ is congruent to triangle $BDE$ and triangle $CEF$. Since corresponding angles in congruent figures have the same measure, I know that angle $CAB$ is congruent to angle $EBD$. Similarly, I know that angle $ABC$ is congruent to angle $BDE$.
2. 72$°$. Explanations vary. Possible response: Translations preserve angle measure so the measure of angle $ABC$ is $63^\circ$ and the measure of angle $EBD$ is $45^\circ$. The measure of angle $CBE$ is $72^\circ$ because $180-63-45=72$.
3. Triangle $BCA$. The corresponding parts of these triangles are congruent so the two triangles are congruent. Because translations preserve side length, $AC$ is congruent to $EB$ and $AB$ is congruent to $EC$. Side $CB$ is part of both triangles. The measure of angle $ACB$ is 72$°$ because the angles in a triangle sum to $180^\circ$. In question 2, the measure of angle $EBC$ is also $72^\circ$. Since angles on a line sum to $180^\circ$, the measure of angle $ECB$ is $63^\circ$ and the measure of angle $BEC$ is $45^\circ$.

Minimal Tier 1 response:

• Work is complete and correct, with complete explanation or justification.
• Sample:

1. Translations preserve angle measure so angle $CAB$ is congruent to angle $FCE$ and angle $CBA$ is congruent to angle $EDB$.
2. $72^\circ$. Because triangle $ABC$ is congruent to triangle $BDE$, the measure of angle $ABC$ is $63^\circ$ and the measure of angle $ECD$ is $45^\circ$. The angles at point C form a line and $180-63-45=72$.
3. Triangle $BCA$. A reflection across the line containing segment $BC$ takes triangle $CBE$ to triangle $BCA$.

Tier 2 response:

• Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
• Sample errors: Explanations in parts a and b appeal to the diagram and are logically sequenced but do not appeal specifically to transformations; good complete explanations but a minor error in calculating angle measures or naming congruent figures; uses the fact that angles sum to $180^\circ$ but does not connect to angles on a line or angles in a triangle.

Tier 3 response:

• Work shows a developing but incomplete conceptual understanding, with significant errors.
• Sample errors: Work does not appeal to transformations or congruent corresponding parts; incomplete explanations and minor errors; three or more error types under Tier 2 response.

Tier 4 response:

• Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
• Sample errors: Little progress on any of the problem parts; justification with many errors or no justification for parts a and c.